Class lanczos (o2scl_linalg)

O2scl : Class List

template<class vec_t = boost::numeric::ublas::vector<double>, class mat_t = boost::numeric::ublas::matrix<double>>
class lanczos

Lanczos diagonalization.

This class approximates the largest eigenvalues of a symmetric matrix.

The vector and matrix types can be any type which provides access via operator[] or operator(), given a suitable vector allocation type.

The tridiagonalization routine was rewritten from the EISPACK routines imtql1.f (but uses std::hypot() instead of pythag.f).

Idea for Future:

The function eigen_tdiag() automatically sorts the eigenvalues, which may not be necessary.

Idea for Future:

Do something better than the simple matrix-vector product. For example, use dgemm() and allow user to specify column or row-major.

Idea for Future:

Rework memory allocation to perform as needed.

Public Functions

inline lanczos()

inline int eigenvalues(size_t size, mat_t &mat, size_t n_iter, vec_t &eigen, vec_t &diag, vec_t &off_diag)

Approximate the largest eigenvalues of a symmetric matrix mat using the Lanczos method.

Given a square matrix mat with size size, this function applies n_iter iterations of the Lanczos algorithm to produce n_iter approximate eigenvalues stored in eigen. As a by-product, this function also partially tridiagonalizes the matrix placing the result in diag and off_diag. Before calling this function, space must have already been allocated for eigen, diag, and off_diag. All three of these arrays must have at least enough space for n_iter elements.

Choosing /c n_iter = size will produce all of the exact eigenvalues and the corresponding tridiagonal matrix, but this may be slower than diagonalizing the matrix directly.

inline int eigen_tdiag(size_t n, vec_t &diag, vec_t &off_diag)

In-place diagonalization of a tridiagonal matrix.

On input, the vectors diag and off_diag should both be vectors of size n. The diagonal entries stored in diag, and the \( n-1 \) off-diagonal entries should be stored in off_diag, starting with off_diag[1]. The value in off_diag[0] is unused. The vector off_diag is destroyed by the computation.

This uses an implict QL method from the EISPACK routine imtql1. The value of ierr from the original Fortran routine is stored in td_lasteval.

Public Members

size_t td_iter

Number of iterations for finding the eigenvalues of the tridiagonal matrix (default 30)

size_t td_lasteval

The index for the last eigenvalue not determined if tridiagonalization fails.

Protected Functions

inline void product(size_t n, mat_t &a, vec_t &w, vec_t &prod)

Simple matrix-vector product.

It is assumed that memory is already allocated for prod.